Finite Group Databasebeta

← #121113

Subgroups — #121113

Order 512.

511 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8112C211yesmaximal, characteristic
2#8108C211yesmaximal, characteristic
3#8057C211yesmaximal, characteristic
4#8016C211yesmaximal, characteristic
5#7983C211yesmaximal, characteristic
6#7982C211yesmaximal, characteristic
7#7915C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1306C414yescomplemented
16#1305—22no—
17#1298—22no—
18#1294—22no—
19#1289—22no—
20#1289—22no—
21#1280—22no—
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—21no—
25C22×M16—21no—
26C23×C8—21no—
27C2×#117—21no—
28C2×#117—21no—
29C22×C4⋊C4—21no—
30C22×C22⋊C4—21no—
31#407—21no—
32#407—21no—
33#407—21no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407C2×C412yes—
42#407C2×C412yes—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8D412yes—
54C2×C22⋊C8Q812yes—
55#381—21no—
56#381C2×C412yes—
57#380—21no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380D412yes—
65#379—21no—
66#379—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C2×#95D412yes—
72C25#4111yescharacteristic
73C23×C4—22no—
74C23×C4—82no—
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×Q811yescharacteristic
80C23×C4#4111yescharacteristic
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—82no—
96C2×M16—82no—
97C2×M16—82no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—82no—
105#117—82no—
106#117—82no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—82no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—82no—
121C2×C42—41no—
122C2×C42—41no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—41no—
145C24—41no—
146C24—42no—
147C24—41no—
148C24—81no—
149C24#11711yescharacteristic
150C24#12311yescharacteristic
151C24#12311yescharacteristic
152C24#12411yescharacteristic
153C24#12511yescharacteristic
154C24#12611yescharacteristic
155C24#12611yescharacteristic
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4#11711yescharacteristic
177C22×C4#11711yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4C4×Q811yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4#12311yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4#12311yescharacteristic
192C22×C4#12411yescharacteristic
193C22×C4#12411yescharacteristic
194C22×C4#12511yescharacteristic
195C22×C4#12511yescharacteristic
196C22×C4#12611yescharacteristic
197C22×C4#12611yescharacteristic
198C22×C4#12611yescharacteristic
199C22×C4#12611yescharacteristic
200C22×C4#12611yescharacteristic
201C22×C4#12611yescharacteristic
202C22×C4#12611yescharacteristic
203C22×C4#12611yescharacteristic
204M16—82no—
205M16—82no—
206M16—82no—
207M16—82no—
208M16—82no—
209M16—82no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—82no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C42—82no—
235C42—82no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—84no—
261C23—84no—
262C23#38311yescharacteristic
263C23#38711yescharacteristic
264C23#38811yescharacteristic
265C23#39711yescharacteristic
266C23#39711yescharacteristic
267C23#40011yescharacteristic
268C23#40011yescharacteristic
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—164no—
307C2×C4#38711yescharacteristic
308C2×C4#38711yescharacteristic
309C2×C4#38811yescharacteristic
310C2×C4#38811yescharacteristic
311C2×C4#39611yescharacteristic
312C2×C4#39711yescharacteristic
313C2×C4#40011yescharacteristic
314C2×C4#40111yescharacteristic
315C8—82no—
316C8—82no—
317C8—82no—
318C8—82no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—81no—
343C22—82no—
344C22—82no—
345C22—81no—
346C22—82no—
347C22—82no—
348C22#144311yescharacteristic
349C22#208012yes—
350C22#208111yescharacteristic
351C22#208211yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—162no—
360C4—162no—
361C4—162no—
362C4#210412yes—
363C2—21no—
364C2—22no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2—41no—
370C2—41no—
371C2#1130111yescharacteristic
372C2#1131811yescharacteristic
373C2#1132011yescharacteristic
subgroupG/H
center, socleC22#1443
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
15. #1306C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#81122
#81082
#80572
#80162
#79832
#79822
#79152